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I've got AMD Radeon HD 7450M graphics card, and it worked perfectly well under Ubuntu 12.10 (64-bit). The same should probably apply to your card as well, which you can test by booting the said computer with a live CD/DVD/USB.

The only problem was the too much power consumption as it was actually a Hybrid Graphics card with an "Integrated" Intel graphics controller and a "Discreet" AMD/ATI one. I finally figured out how to turn off the latter when the computer is running on battery power. See: Error Creating .deb File of AMD Catalyst Driver